# Questions on the proof of Cayley Hamilton Theorem

In the proof presented in my textbook, it utilises the equation

$$(A-\lambda I) \operatorname{Adj}(A - \lambda I) = \det(A-\lambda I) \, I$$

And it stated that,

$$P(A-\lambda I) = \det(A-\lambda I)= a_0 + a_1\lambda + ... + a_n\lambda^n$$

and that

$$Q(A-\lambda I) = \operatorname{Adj}((A-\lambda I) = b_0 + b_1\lambda + ... + b_k\lambda^k$$

I have no problem understanding that $$P(A-\lambda I)$$ is the characteristic polynomial for matrix $$A$$. However, I have problem understanding 2 things,

1. Why $$\operatorname{Adj}(A-\lambda I)$$ can be written into a polynomial?

2. What does it mean to then multiply the above polynomial to matrix $$(A-\lambda I)$$?

Many thanks!

• You can answer your questions for $n=2$ yourself as follows. Write out all expressions explicitly for $A=\begin{pmatrix} a & b \cr c & d \end{pmatrix}$. How do you define $Adj$? You can compute it explicitly for $A$, see here. – Dietrich Burde Mar 2 at 9:25
• I mean if I just do the operation $\begin{pmatrix} d-\lambda & -b \cr -c & a-\lambda \end{pmatrix}$ $\begin{pmatrix} a-\lambda & b \cr c & d-\lambda \end{pmatrix}$. What I get is $\begin{pmatrix} (a-\lambda)(d- \lambda) - bc & 0 \cr 0 & (a-\lambda)(d-\lambda) + bc \end{pmatrix}$. Correct? – Klein_Bottle Mar 2 at 11:08
• Adj$(A-\lambda I)$ can be thought of as a polynomial in $\lambda$ whose coefficients are matrices. I think Adj$(A-\lambda I)=B_0+B_1\lambda+\cdots +B_{n-1}\lambda^{n-1}$ is a better way of writing it. – Lord Shark the Unknown Mar 2 at 11:11
• The equality in your question should be $\color{red}{(A-\lambda I)}\operatorname{Adj}(A-\lambda I)=\det(A-\lambda I)I$, not $\color{red}{A}\operatorname{Adj}(A-\lambda I)=\det(A-\lambda I)I$. – user1551 Mar 2 at 13:20

If we want to evaluate the polynomial $$\text{det}(\phi - t I)$$ at a matrix, it is natural to view it as an element of $$\text{End}_{\mathbb{F}} (V) [t]$$. The factorization $$\text{det}(t I - \phi ) = \text{adj}(tI - \phi ) (tI - \phi)$$ makes sense if we allow ourselves to think of $$\text{adj}(tI - \phi)$$ as a polynomial with endomorphisms as entries.
After we think of these as elements in $$\text{End}_{\mathbb{F}} (V)[t]$$, the product $$\text{adj}(A - \lambda I) ( A - \lambda I)$$ works just like any polynomial ring over a noncommutative ring:
$$\sum_{i = 1}^n \phi_i t^i \sum_{j = 1}^m \psi_j t^j = \sum_{i = 1}^{n + m} \sum_{j + k = i, 1 \leq j \leq n, 1 \leq k \leq m} \phi_j \circ \psi_k t^{i}$$
In fact, this language is convenient for the proof: we have an isomorphism $$\text{End}_{\mathbb{F}} (V) [t] \cong \text{End}_{\mathbb{F}} (V \otimes_{\mathbb{F}} \mathbb{F}[t])$$ In $$\text{End}_{\mathbb{F}} (V \otimes_{\mathbb{F}} \mathbb{F}[t])$$, we have a factorization $$\text{det}(\mu_t - \phi) 1_V = \text{adj}(\mu_t - \phi) (\mu_t - \phi)$$ Where $$\mu_t$$ is multiplication by $$t$$. Under the isomorphism, this becomes a factorization $$p(t) = f(t)(t - \phi)$$ where $$p(t) \in \text{End}_{\mathbb{F}} (V)[t]$$ is the characteristic polynomial.